Coupled quintessence scalar field model in light of observational datasets

We do a detailed analysis of a well-theoretically motivated interacting dark energy scalar field model with a time-varying interaction term. Using current cosmological datasets from CMB, BAO, Type Ia Supernova, H(z) measurements from cosmic chronometers, angular diameter measurements from Megamasers, growth measurements, and local SH0ES measurements, we found that dark energy component may act differently than a cosmological constant at early times. The observational data also does not disfavor a small interaction between dark energy and dark matter at late times. When using all these datasets in combination, our value of H 0 agrees well with SH0ES results but in 2.5σ tension with Planck results. We also did AIC and BIC analysis, and we found that the cosmological data prefer coupled quintessence model over ΛCDM, although the chi-square per number of degrees of freedom test prefers the latter.


Introduction
The significant observational evidence about Universe's dynamics [1][2][3] hint at the accelerated expansion at present.The elementary candidate to reveal the nature of this new physics includes Einstein's cosmological constant 'Λ' with the equation of state (EoS) 'ω' = −1 in the ΛCDM model.Though this model provides a good fit to the latest observations, it lacks solutions to other significant problems [4] in cosmology, especially to the fine-tuning problem and cosmic coincidence problem, also known as famous Why now? problem.Along with this, we still have no information about the nature of Dark Energy and Dark Matter.All we know is that our universe is composed of two types of perfect fluids, one of which is baryonic and responsible for the deceleration of the universe and the growth of structures.The other one, dominant at late times, has negative pressure and is responsible for the acceleration of the present universe.Both the components have different equation of state parameter ω = P /ρ.The baryonic component is known to be like radiation-dominated, stiff matter, or the dustlike universe.However, the determination of the equation of state for dark energy (ω DE ) is still one of the important unresolved questions in cosmology.Recent studies [5][6][7][8][9] rules out the possibility of ω DE << −1 and other studies, SDSS and WMAP, [10] and [11] put bounds as −1.67 < ω DE < −0.62 and −1.33 < ω DE < −0.79 at present respectively.
Very recent results [12] and [13] very clearly rule out the possibility of rigid Λ by 5σ.In a complete cosmology model-independent way, the authors [14] showed at present, the universe requires a phantom equation of state (ω DE < −1) to explain the present observational datasets.
Thus, probing the cosmologists to explore beyond the standard model for dark energy candidates, perhaps dynamical dark energy model with varying EoS (ω), see [15][16][17][18], and references therein.The simplest possibility consists of a light scalar field such as a quintessence field, varying slowly during the Hubble time and can drive the accelerated expansion [19][20][21][22][23][24].Some studies also argue that the quintessence scalar field should couple (non-gravitationally) with the matter as provided by [25] unless some special symmetry inhibits the coupling.Such a coupling or interaction between Dark Energy (DE)/quintessence scalar field and Dark Matter (DM) can be characterized by a continuous energy flow or momenta between the two components.Many studies in the literature prescribe a purely phenomenological approach to deal with interacting cosmologies of dark sectors.Then, other classes of models are motivated from a field theory perspective and are well motivated.Numerous observations [26][27][28][29][30][31][32][33][34][35][36][37] are in support of the theory of dark sector interaction, constraining energy transfer from DM to DE or the opposite paved their way into existence.Cosmological observations indicate that the coupling impacts the universe's evolution [38,39].Effects of interaction on the CMB and linear matter power spectrum [40,41] (for detailed review [42]), the structure formation at different scales and times and halo mass function [43][44][45][46][47] and on the behavior of the cosmological parameters [48][49][50][51] have been investigated in respective works.Based on these, coupled quintessence models were also studied theoretically in several other contexts [52][53][54][55][56][57] to provide a viable cosmological solution.Interacting Dark Energy Models have also been studied extensively to alleviate "Hubble Tension" -which is now confirmed to be more than 5σ after SH0ES 2022 results [13,[58][59][60].Along with Hubble Tension is coupled the S 8 tension or r d r ag tension, i.e., the sound horizon at the drag epoch tension because BAO and CMB measure the product of H 0 and r d r ag and not r d r ag alone [61,62].Several works [34,42,[63][64][65][66][67][68][69][70][71][72][73][74][75][76][77] tried to address these tensions in cosmology but could only solve one tension or the other by keeping the same number of parameters.The only solutions [65,[78][79][80][81][82][83][84][85] to resolve all cosmological tensions at present are those which are extremely fine-tuned or have more parameters and therefore increasing the uncertainties in the parameter space.
In the present work, we do a detailed and elaborate study of late-time cosmological dynamics.We mainly study the well-motivated dark matter and dark energy interaction case theoretically and try to understand how current observational data sets like CMB, Type Ia Supernovae, cosmic chronometers, BAO, Masers, growth rate data, and local H 0 measurement from SH0ES study influences the constraints on different cosmological parameters.
The work is structured as follows.In the next section, we present the model framework for the coupled dark sector and discuss its dynamics in brief.Later, we work out the linear dark matter perturbation evolution equation in section 3. Section 4 describes the observational data and methodology adopted in this work.Section 5 presents the results and thorough discussion from the observational analysis.We conclude the findings in Section 6.

Coupled field model and the Background dynamics
In our study, we assume that the geometry of the universe describing flat, isotropic, and homogeneous universe is given by Friedmann-Lema î tre-Robertson-Walker (FLRW) metric and the line element for it is given by ds which when transformed into spherical coordinates reads as (2.2) where k = −1, 0, and 1 denotes open, flat, and spatially closed universe.We consider a canonical scalar field φ (as dark energy component) and non-relativistic (cold) dark matter component with pressure P d m = 0 undergoing an interaction that leads to a coupled quintessence dark energy model.Under this assumption, the action for dark sector coupling can be set up as L m is the matter Lagrangian for different matter field species, which also depends on the field φ through the coupling.Different matter species (i ) may experience different couplings [86,87].However, radiation is assumed uncoupled to dark energy because of conformal invariance.We also ne-glect the baryons since coupling to the baryons is strongly constrained by the local gravity measurements [88].Thus, our study assumes that only dark matter couples to the dark energy scalar field.By taking varying the action (2.3) w.r.t. the inverse metric, we get Einstein's gravitational field equation as follows where T µν is the sum of T (φ) µν (energy-momentum tensor of DE component) and T (m) µν (energy-momentum tensor of matter component).Since we allow interaction between the dark species, they do not evolve independently but coupled.Furthermore, they satisfy the local conservation equation in the form given: where Q ν expresses the interaction between dark matter and dark energy.
in which F ,φ ≡ ∂F /∂φ.F (φ) = F 0 exp(βφ) is the coupling strength.β is a constant.The radiation component, on the other hand, being not interactive with dark species, is separately conserved and follows ∇ µ T (r ) µν = 0. Using the Einstein equations, Friedmann equations for the coupled case scenario can be expressed as From the conservation of energy-momentum tensor (2.6) or equivalently from equation (2.7), one can easily derive energy conservation or continuity equation for each component. ρφ where H ≡ ȧ a , indicates the time evolution of the universe.The dot indicates derivative with respect to time t , and ρ ′ d m = F (φ)ρ d m denotes the energy density of dark matter.Corresponding time component of interaction term in Eq.(2.6) is given by Q = F ,φ ρ d m φ.
To study the dynamics of cosmological scalar fields in the presence of background fluid, it is easier to work with dimensionless variables.In our case, we introduce (2.9) along with where 'prime' denotes derivative w.r.t.ln(a).And we define energy fraction for dark energy as and dark matter energy density as satisfying Ω φ + Ω d m + Ω r = 1.We introduce one another dimensionless quantity 'γ' for describing dark energy equation of state 'ω φ ' as: x 2 + y 2 .
(2.11) Also, total (effective) equation of state parameter 'ω e f f ' becomes 12) The coupled dynamical set of equations can now be expressed as: , (2.13) where which for potential V (φ) ∝ e −λφ (λ = constant) and for coupling parameter F (φ) = F 0 exp(βφ) gives unity.To solve the above autonomous system set, we assume the initial value of ω φ to be near -1, i.e., γ ≃ 0.0001 [24].Other quantities, namely Ω φ , Ω r , λ, and m are free parameters.Now, we study the dynamics of the system (2.13) with the stationary points which satisfy the constraint (2.10).The physically viable critical points (x c , y c , Ω d m c ) are Point P 1 explains the radiation-dominated universe as seen from Eq.(2.10) and takes the value of effective EoS parameter ω e f f as 1/3.Points P (±)  2 describe the universe dominated by the kinetic part of the scalar field.Thus the solution corresponds to stiff fluid with the effective equation of state parameter ω e f f = ω φ = 1.Moreover, stationary points P (±) 3 describe an accelerated universe when | λ |< 2. A corresponding de Sitter solution occurs only when λ = 0 making ω e f f = −1, i.e., the field potential plays the part of the cosmological constant.Points P (±)  3 are physically acceptable under the constraint | λ |< 6.However, points P (±)  4 produces a dark energy dominated accelerated solution when | λ |< (2 2 − 1).
We now study the stability of the above stationary points.The following are the eigenvalues corresponding to the stationary points From the eigenvalues, we find that the point P 1 and the point P (±)  2 exhibit saddle nature.Moreover, for m = λ = 0, point P (±)  2 exhibits instability too.Point P (±) 3 becomes a stable attractor for λ 2 < 3 and m < 3 λ − λ and it is a repeller when λ 2 > 6 and m > 3 λ − λ.Otherwise, the point shows saddle behaviour.Furthermore, point P (±)  4 acts as an attractor for m < 1 4λ (3 + λ 2 ) and n < s and becomes unstable when m > 1 4λ (3 + λ 2 ) and n > − s.Point P (±) 4 can also describe an oscillatory solution if 's' value becomes negative.
With these results, we now continue to investigate the perturbation theory for this model and its implications when confronted with cosmological observations.

Linear Perturbation Equation
Since the coupling modifies the evolution of matter perturbations and the clustering properties of galaxies, to gain further insight, we now study the evolution of the matter perturbations and, thence, include the linear growth rate data ( f σ 8 ) from large-scale structure (LSS) survey.We re-scale the term "F (φ)ρ d m " as ρ c .Given this, we define the perturbation variable as δ c ≡ δρ c ρ c , where the subscript c stands for the (cold) dark matter component and δρ c is the deviation from the background dark matter density.Mathematically, the equation governing the growth of perturbation (in the Newtonian gauge) at linear regime is given as The introduction of coupling to the dark matter then modify the growth of matter perturbation as which can also be written in terms of dimensionless variables as We write the modified Hubble normalized, in terms of N = ln a as The dark matter perturbations and the Hubble evolution for the uncoupled case are reproduced by setting m = 0. We consider the time evolution of the universe at N i = −7 to ensure the evolution around the matter-dominated epoch.Therefore, we use the initial conditions as φ( And for the matter density contrast we take The dynamical set of equations (Eq.2.13) and the perturbation equation (Eq.3.3) are addressed with the cosmological observations to constrain the interacting scalar field quintessence model and its parameters.

Observational data
We devote this section to presenting and describing the main observational data implemented to constrain the model parameters and the statistical analysis results.
• The Cosmic Microwave Background (CMB) data are an effective probe for observational analysis of the cosmological models.In this work, we work with the CMB distance priors on the acoustic scale l A leading to the alteration of the peak spacing and on the shift parameter 'R' affecting the heights of the peaks.We consider using the Planck 2018 compressed likelihood TT, TE, EE + lowE obtained by Chen et al [89] (see Ade et al. [90] for the detailed method for obtaining the compressed likelihood).
• For SN data, we use the SNe Ia samples of 1701 light curves of 1550 distinct Type Ia supernovae (SNe Ia) in the redshift range z ∈ [0.001, 2.26] from the latest Pantheon+ compilation [91,92].This sample also includes SH0ES Cepheid host distance anchors (Riess et al, denoted as R21 [93]).Thus, for H 0 , when combining it with Pantheon+, we use the PantheonPlusSH0ES likelihood (refer Eq. 15 of [92]), where the Cepheid calibrated host-galaxy distance provided by SH0ES facilitates constraints on H 0 .
• The cosmic chronometers (CC) approach allows us to obtain observational values of the Hubble function at different redshifts z ≤ 2 directly.Since these measurements are independent of any cosmological model and Cepheid distance scale, they can be used to place better constraints on it.In the present analysis, we measure H (z) using the CC covariance matrix [94][95][96].
• The H 2 O Megamaser (hereafter, "MASERS") technique under the Megamaser Cosmology Project (MCP) leads to the direct measurement of the H 0 by measuring angular-diameter distances to galaxies UGC 3789, NGC 5765b, and NGC 4258 in the Hubble flow redshifts z = 0.0116, 0.0340, and 0.0277 respectively.[102][103][104][105][106]. The technique is based only on geometry, independent of standard candles and the extra-galactic distance ladder, and may provide an accurate determination of H 0 .
• In addition to geometric probes, we use the growth rate ( f σ 8 (z)) data provided by various galaxy surveys as collected in [107] to constrain cosmological parameters.Now, to constrain the free and derived parameters of this coupled cosmological scenario, we use the Markov Chain Monte Carlo (MCMC) technique, emcee: the MCMC Hammer [108].We work with the following parameter space P ≡ Ω φ i * 10 −9 , r d r ag , λ i , m i , Ω r i , h, σ 8 during the statistical analyses and the priors employed on these cosmological parameters are enlisted in Table 1.The physical limits are intact where Ω φ > 0, Ω d m > 0 and Ω r > 0. During our further analysis, the Hubble constant is assumed to be H 0 = 100h km s −1 Mpc −1 ; hence, we define the dimensionless parameter h.We have fixed the baryon density parameter, Ω b 0 to be 0.045 according to CMB constraints from Planck (2018) [58], which is also in agreement with Big Bang Nucleosynthesis (BBN) constraints and n s to be 0.96.One should note that we have chosen positive priors on the interaction parameter.However, the results do not change if we choose negative or bigger priors alternatively.

Results and Discussion
We start by discussing the results obtained within the interacting canonical context.In table 2, we present the mean values and the 1σ ranges of different cosmological parameters with both coupled and uncoupled scenarios.To see the effect of the inclusion of each data set in the coupled scenario on the constraint of each parameter, we refer to Fig. 1, where we show 1-dimensional marginalized posterior distributions and the corresponding 2D contour plots at 68% and 95% confidence level (CL) for all independent parameters.We gradually added PantheonPlus, f σ 8 , and H 0 to the data set combination BAO + Masers + CMB + CC (further renamed as BASE combination) and can see more constrained parameters by adding more datasets.We also see a similar effect in the non-coupling case, as shown in Fig. 2. Since constraints on parameters show not much difference in the case of coupled or un-coupled scenarios, we, from now onward, focus on further studying the coupled case.Figure 1: 68% and 95% CL contour plots and corresponding one-dimensional marginalized posterior distribution for all cosmological parameters obtained from the MCMC analysis within the present coupling scenario utilizing several combinations of data sets.

The H 0 and σ 8 Plane
We see that 1σ confidence level contours are relatively larger for CMB + CC + BAO + Masers (BASE) data than other data combinations.Whereas the addition of PantheonPlus sample significantly reduces the constraint on parameter Ω d m , with the difference of ∼ 0.1σ to the BASE combination.This alteration estimates slightly higher uncertainty in r d r ag , differing by less than 0.1σ relative to the BASE data.We also observe the change in the λ i parameter with less than 0.3σ significance compared to the BASE combination, refer Table 2. Furthermore, the inclusion of f σ 8 and H 0 data to the previous data combination strongly reduces the parameter space on σ 8 and h, respectively, as seen in Fig. 1.
Except parameter h and parameter σ 8 , we also find the notable impact on other parameters such as r d r ag and λ i when we add SH0ES data or when we include f σ 8 data in the analysis.
In this study, we found that CMB + CC + BAO + Masers (BASE) data combination shows H 0 = 67.1±2.8 km/s/Mpc at 68% CL which is compatible well within 0.3σ range with H 0 measurement Advancing with S 8 = σ 8 Ω d m 0 /0.3, the addition of f σ 8 measurement yields systematically low value of S 8 and resulted in S 8 = 0.758±0.023raising the difference with Planck (2018)+ΛCDM (S 8 = 0.830 ± 0.013) to precisely at 2σ.This prediction of S 8 is in excellent agreement with the KiDS-1000-BOSS results S 8 = 0.766 +0.020 −0.014 [109], differing by 0.3σ.Whereas the estimation is in good agreement with DES-Y3 results S 8 = 0.776±0.017[110] with the difference of 0.5σ.The results from CMB + CC + BAO + Masers + PantheonPlus + f σ 8 (z) are more robust in tightening constraints on the σ 8 or the S 8 parameter and justify the reduced uncertainty and the parameter space compared to the results from without growth rate data combination; see Table 2 and Fig. 1.Finally, the combined data constraint yields S 8 = 0.757±0.023,a difference of 2.5σ from the results of Planck (2018)+ΛCDM [58].However, this analysis of combined data produces essentially the same results as in the previous analysis maintaining a good agreement of 0.3σ and 0.5σ with the measurements obtained from KiDS-1000-BOSS survey [109] and DES-Y3 survey [110], respectively.
To assess the robustness of our findings and to check for the ambiguity concerning the prior dependency in the obtained results we present the results with different priors on the cosmological parameters in Table 5 and in Table 7, in the Appendix.We found that the values of the key cosmological parameters (both mean and error bars) are independent of priors only when the f σ 8 is in consideration along with other data sets of CMB, SNIa, CC, BAO, MASERS, and H 0 .This is explained in detail in the Appendix.

The h-r d r ag Plane
In addition to a discrepancy in the Hubble constant, there is a discrepancy in the co-moving sound horizon at the end of the baryon drag epoch, r d r ag as well.The two parameters H 0 and r d r ag are strictly related when we consider BAO observations.In actuality, a combination of expansion history probes such as BAO and PantheonPlus data can provide a model-independent estimate of the lowredshift standard ruler, constraining the product of h (with H 0 = h × 100 km/s/Mpc) and the sound horizon r d r ag directly.This implies that, to have a higher H 0 value in agreement with SH0ES, we need r d r ag ∼ 137 Mpc, while to agree with Planck, assuming ΛCDM, we need r d r ag ∼ 147 Mpc.For this reason, the solutions that increase the expansion rate and at the same time decrease r d r ag are most promising.In our analysis, this feature is completely in agreement with the Planck (2018) estimates due to the strong compatibility of estimated H 0 with the one from Planck (2018) + ΛCDM model.However, the addition of SH0ES result gives H 0 = 71.7±1.54 km/s/Mpc, that is closer to the R21 value of H 0 , also leads to an empirical determination of r d r ag near 139 Mpc.This modest discrepancy in the sound horizon value r d r ag from the one in [111] agrees with the slight discrepancy in the H 0 measurements from SH0ES.This correlation can also be seen from table 2. Evidently, this cosmological solution is promising and consistent with the fact that the relation hr d r ag is constant by the BAO measurements.

The S 8 -Ω d m Plane
The model that allows larger H 0 values tend to introduce other tensions, such as higher S 8 values.
To obtain simultaneously higher values of H 0 , lower values of S 8 , and consistent values of Ω d m is also necessary to define the correct evolution.Since adding the BAO and SNe measurements to the Planck data strengthens the constraints towards Planck values, the correlation between their combined results is relatively strong.Referring to the Table 2, f σ 8 addition improves the S 8 consistency with KiDS-1000-BOSS [109] and DES-Y3 [110] results.This is also illustrated in Fig. 4. We, particularly, show that only dataset that includes f σ 8 , constrain S 8 or σ 8 better while other datasets (without f σ 8 ) fails to put constrain on it.From this joint analysis, we found a lower and a well-constrained estimation for S 8 , leading to a lower and a constrained value for Ω d m (Table 2 and Fig. 4).Moreover, using the full combined likelihood for the data considered in this work we found the similar results with further increment in the H 0 estimated value while S 8 retaining the consistency with KiDS-1000-BOSS and DES-Y3 survey results.

The Co-moving Hubble Parameter
The recent BAO measurements along the line of sight and transverse directions lead to joint constraints on H (z)r d r ag .Since Planck + ΛCDM constrains r d r ag to a precision of 0.2 %, the BAO measurements can be accurately converted into absolute measurements of H (z).These constraints on H (z) from the BOSS analyses are plotted in Fig. 5.The error bars are constraints from BOSS DR12 galaxy sample [101], eBOSS DR14 quasar sample [99], the correlations of Lyα absorption in eBOSS DR14 at z = 2.34 [112] and from the Lyα auto-correlation and cross-correlation with quasars from SDSS data release DR12 [100].The error bar at z = 0 shows the inferred distance-ladder Hubble measurement from R21 [93].
The illustration in Fig. 5 shows clearly how well the dark energy interaction model fits the BAO measurements of the Hubble parameter except for the DR12 Ly-α data point.It is also consistent with the R21 measurements of Hubble at z = 0 for all the data sets combined in this work within 1σ error bars.This is also illustrated in Fig. 6, which shows the combined constraints on h and Ω d m from different data combinations.Without adding H 0 to the combination of CMB, BAO, Pantheon-Plus, H (z), Masers, and f σ 8 , the tension with SH0ES measurement of H 0 still prevails as seen in table 2, though the same results show lower mean value and well improved constraints on Ω d m as compared to the BASE data.However, the addition of SH0ES data (green contour) constrain Ω d m towards slightly lower mean value and shifts H 0 closer to the R21 measurement.

Other Cosmological Parameters
We further analyse the model for completeness' sake and clarity.We have plotted the energy density evolution parameter for dark energy and dark matter, along with 68% and 95% CL for the coupled case using all the data presented in this work in Fig. 7. Fig. 8 shows the overlapping confidence contours for the equation of state parameter as a function of redshift from coupled and uncoupled models using all the data presented in this work.The solid yellow and red lines indicate the best-fit values for the DE EoS parameter for coupled and uncoupled models, respectively.However, Fig. 9 shows the Ω dm(c) Figure 9: The best-fit evolution of key mological parameters for the coupled and uncoupled models is shown for the combined set of CMB, SNIa, CC, BAO, Masers, H 0 , and growth data.The solid lines and dashed-dot lines show the evolution in the coupled case and un-coupled case, respectively.
density parameters and equation of state parameters evolution from their best-fit values as a function of the logarithmic scale factor.These analyses exactly show the expected behaviour of the observed universe at present, which is dark energy-dominated accelerated expansion.Moreover, while fitting with the data, we found that the model favors small values of the coupling parameter "Q", making the parameter evolution indistinguishable compared to the one from the uncoupled case.However, these predictions are model-dependent and may vary for other coupling models.

Comparison with ΛCDM results
In Table 3, we present the constraints that we got from χ 2 minimization fitting assuming the standard ΛCDM model.We also show how the mean values change within the same model in addition to different datasets.We observe the notable impact on the addition of PantheonPlus sample to the BASE combination that significantly changes the parameter, Ω d m values (both mean and error bars).These changes narrow down the Ω d m parameter space.Growth rate ( f σ 8 (z)) data addition reflects more precised constraint on σ 8 or S 8 parameter compared to "CMB + PantheonPlus + CC + BAO + Masers" data set, keeping other parameters unaltered.Further inclusion of local H 0 value from SH0ES shifts the mean value of h towards Riess et al. value [13] with smaller error bars, differing by ∼ 1.5σ relative to the h value obtained from without SH0ES data sets.The effect of the shift in h can also be seen in r d r ag parameter values (both mean and reduced error bars) and in Ω d m parameter value with lower mean.

Model Selection Statistics
In this section, we discuss the comparison of our models (coupled and uncoupled) with the standard ΛCDM model.mi n value for the coupled model when compared to the un-coupled or ΛCDM model.For all the data together, this leads to a large reduction in AIC and BIC estimates for the coupled model, and hence, the same is significantly preferred over the ΛCDM model.
We also considered χ 2 mi n per number of degrees of freedom approach for the present model, aiming to understand the observational solidity of the model with respect to the standard reference model.We found that χ 2 mi n n d o f raises an over-fitting issue with the coupled model opposing the analysis from AIC.This trait can be found in Table 4.However, using n d o f analysis, ΛCDM is still a preferred candidate but from the perspective of AIC and BIC analyses, coupled cosmology is the preferred scenario for the universe's evolution.

Final Remarks
In this work, we study an interacting dark energy-dark matter model having time-varying interaction term Q = F, φ ρ d m φ.This work is interesting because it is theoretically very well motivated from a field theory perspective, not just phenomenological.We can see that because of the time-varying interaction term, the dark energy component may differ from the cosmological constant at very early times (z = 1100).Also, a small interaction is not disfavoured by the data at late times.However, we observed that there is no significant difference in the constraining power of the coupled model over the uncoupled one.We found that the parameter m i which is the coupling parameter peaks at the very lower value in the prior range provided, which makes the coupling significantly low making it indistinguishable from the uncoupled case.
Initially, we tested our model to find the future attractor solution by formulating a coupled dynamical set of equations.Later, we subjected the model to various data combinations and tried to understand better the influence of each cosmological data set on respective cosmological parameters.
We found that while using all the datasets together, we found H 0 value to be in excellent agreement with the latest local determination of H 0 by R21 [93], though in substantial (∼ 2.5σ) tension with Planck [58] measurements.It is widely discussed that the models that alleviate the H 0 discrepancy do not necessarily solve other tensions, such as a higher value of σ 8 or S 8 .Similarly, in our work, we found that the coupled model provides a remarkably better fit to the CMB, CC, BAO, Masers, PantheonPlus, f σ 8 and H 0 in combination with each other and predicts a (lower) late-time structure growth parameter S 8 = 0.757±0.023.The S 8 value is in good agreement with KiDS-1000-BOSS and DES-Y3 estimations, although in moderate tension with Planck 2018 + ΛCDM results which prefer a roughly 2.5σ higher value of S 8 compared to our model prediction.
We also emphasize the parameters like the sound horizon at the drag epoch r d r ag and the dark matter energy density Ω d m , and their respective correlations, in terms of h-r d r ag and S 8 -Ω d m planes.The findings predicted the smaller value of r d r ag for a larger value of H 0 at the cost of increasing disagreement with the Planck data estimates.Moreover, the results from the S 8 -Ω d m parameter space showed marginal tension with Planck with lower and constrained values of S 8 and Ω d m for all the combined data in this work.The illustration in Fig. 5 clearly showed some disagreement (∼3σ) with high-redshift BAO measurements from quasar Lyα observations.The interacting model confirms the overall good consistency of reconstructed H (z) with combined BAO measurements of H (z). Later, we also presented the time evolution of key cosmological parameters along with their confidence ranges for the coupled cosmological model.We found no significant distinction in the parameter evolution compared with the uncoupled scenario due to very low coupling parameter values preferred by the cosmological datasets.Later, we also presented the time evolution of density parameters, the equation of state parameters, and their two-sigma confidence region for the coupled cosmological model.
We then discussed the model comparison techniques to measure the goodness of fit of a given model to the data.We found that introducing extra parameters improves the fit determined by AIC and BIC and strongly favors a complex, coupled model over the ΛCDM one.Moreover, this model introduces two extra parameters, so the "weighted" chi-square test favors the ΛCDM for all the combined data.Such inconsistencies and discrepancies motivate us to look further for the best theoretical model to describe the universe.Following the above analyses, these differences could signify either the need for new exotic dark energy physics to withstand the data or new observations to improve the quality of the data.
In the near future, we shall extend our analysis to incorporate one extra element which can mimic phantom dark energy at present, along with the already present interacting dark matter-dark energy scenarios to resolve present cosmological tensions.

Figure 5 :
Figure 5: Co-moving Hubble parameter as a function of redshift, clearly showing the onset of acceleration just before the redshift around z = 0.6, reconstructed using a combination of CMB, SNIa, CC, BAO, Masers, H 0 and growth data for interacting model.The solid blue line shows the best-fit reconstructed results, while the outer regions show results within 1σ, 2σ, 3σ CL separated by orange, pink and red-dashed lines. 0

Figure 7 :Figure 8 :
Figure 7: The 68% and 95% CL along with the mean value for relative energy density parameters Ω d m (cyan) and Ω φ (purple) for the coupled model with respect to redshift (z).The evolution is plotted for all the data combined together, considered in this work.The blackdashed and blue-dashed lines corresponds to the best-fit values of DM and DE density parameters.
f method, it is clear that the ΛCDM model has the goodness of fit (GoF) to the data and hence, is the favoured model.In summary, from the point of view of the χ 2 mi n

Table 1 :
Flat priors on various parameters of the coupled model.

Table 2 :
Mean values with 68% confidence level (CL) errors on cosmological and free parameters within the interacting and non-interacting paradigm from various data combinations.

Table 3 :
Ω d m 0 /0.3.As evident from Table2, the σ 8 measurements from growth rate ( f σ 8 ) data, along Mean values with 68% confidence level (CL) errors on cosmological and free parameters within the ΛCDM paradigm from various data combinations.withCMB, SNIa, CC, BAO, MASERS, and H 0 data, are well constrained as compared to datasets that do not include f σ 8 , i.e., "CMB + CC + BAO + MASERS" and "CMB + PantheonPlus + CC + BAO + MASERS" combinations.This discrepancy in σ 8 significantly impacts the reliability of the constraints on parameter S 8 for these data combinations (without f σ 8 ) because obtained S 8 solution arises from the parameter being unconstrained.Due to this, we refrain to present the constraints placed on S 8 for these combinations.
[93]/Mpc, is consistent within < 1σ (mildly greater than 0.5σ) range with the measurement of the Hubble constant, H 0 = 73.04 ± 1.01 km/s/Mpc derived from the baseline fit with measurements from the latest SH0ES analysis in R21[93].We now present our multi-probe constraints on the structure growth parameter S 8 , using S 8 = σ 8 The CMB + SNIa + CC + BAO + Masers + f σ 8 + H 0 data constraints on h and Ω d m in the coupled model, compared to the results from without H 0 data.Adding SH0ES H 0 constrains the Hubble parameter, comparable to that R21 estimation.Darker and lighter contours show 68% and 95% of the probability, respectively.
The ΛCDM model corresponds to the dark energy equation of state -1, whereas in our model, the dark energy equation of state is allowed to vary.One way to compare these dark energy models is Akaike Information Criterion (AIC), where AIC is defined as AIC = χ 2 mi n + 2 • d.Another way is Bayesian Information Criterion (BIC), where BIC is defined as BIC = χ 2 mi n + d • ln(N ).χ 2 mi n is the chi-squared minimization to measure how well the data fit a model.'d' is the number of parameters in the model and 'N ' is the number of observational data points.The lowest chi-squared and lowest AIC or BIC at each point indicate where the parameters and model most closely match the measured data.Comparing both the methods, we can see from table 4 that there is always an improvement in χ 2

Table 4 :
The χ 2 mi n , AIC and BIC estimation for coupled/un-coupled and ΛCDM model for the combination of all data set.